# MIT Probability Seminar

## Current Organizers

• Alexei Borodin
• Promit Ghosal
• Jimmy He
• Elchanan Mossel
• Philippe Rigollet
• Scott Sheffield
• Yair Shenfeld
• Nike Sun
• Dan Mikulincer
• Subscription to a mailing list

# Fall 2022

Monday 4.15 - 5.15 pm

Room 2-147

Scheduled virtual talks will be held on Zoom, Monday 4:15-5:15 pm.
A link to a Zoom classroom will appear here!!

## Schedule

• September 12

Room: 2-147

Guilherme Silva
Universidade de São Paulo (ICMC - USP)

Universality for a class of statistics of Hermitian random matrices and the integro-differential Painlevé II equation.

Abstract:

It has been known since the 1990s that fluctuations of eigenvalues of random matrices, when appropriately scaled and in the sense of one-point distribution, converge to the Airy2 point process in the large matrix limit. In turn, the latter can be described by the celebrated Tracy-Widom distribution.

In this talk we discuss recent findings of Ghosal and myself, showing that certain statistics of eigenvalues also converge universality to appropriate statistics of the Airy2 point process, interpolating between a hard and soft edge of eigenvalues. Such found statistics connect also to the integro-differential Painlevé II equation, in analogy with the celebrated Tracy-Widom connection between Painlevé II and the Airy2 process.

• September 19

*** Special Seminar Starting at 3pm! ***

Room: 2-132

Matteo Mucciconi University of Warwick

A bijective approach to solvable KPZ models

Abstract:

Explicit solutions of random growth models in the KPZ universality class have attracted, in the last two decades, significant attention in Mathematical Physics. A common approach to the problem, explored in the last 15 years, leverages remarkable relations between the KPZ equation and quantum integrable systems.

Here, I will introduce a new approach to the solutions of KPZ models, based on a bijection discovered by Imamura, Sasamoto and myself last year. This is a generalization of the celebrated Robinson-Schensted-Knuth correspondence relating at once 1) solvable growth models, 2) determinantal point processes of free fermionic origin and 3) models of Last Passage Percolation on a cylinder.

I will enumerate some of the early applications of this new approach and I will give an overview of the technical tools needed, that include Kashiwara's crystals or the inverse scattering method for solitonic systems.

• September 26

Room: 2-147

Abstract:

• October 3

Room: 2-147

Learning low-degree functions on the discrete hypercube

Abstract: Let f be an unknown function on the n-dimensional discrete hypercube. How many values of f do we need in order to approximately reconstruct the function? In this talk we shall discuss the random query model for this fundamental problem from computational learning theory. We will explain a newly discovered connection with a family of polynomial inequalities going back to Littlewood (1930) which will in turn allow us to derive sharper estimates for the the query complexity of this model, exponentially improving those which follow from the classical Low-Degree Algorithm of Linial, Mansour and Nisan (1989). Time permitting, we will also show a matching information-theoretic lower bound. Based on joint works with Paata Ivanisvili (UC Irvine) and Lauritz Streck (Cambridge).

• October 10

Room: 2-147

National Indigenous Peoples Day.

Abstract: Indigenous Peoples' Day is a holiday in the United States that celebrates and honors Native American peoples and commemorates their histories and cultures. On October 8, 2021, U.S. President Joe Biden became the first U.S. President to formally recognize the holiday,[2] by signing a presidential proclamation declaring October 11, 2021, to be a national holiday.[3] It is celebrated across the United States on the second Monday in October, and is an official city and state holiday in various localities. It began as a counter-celebration held on the same day as the U.S. federal holiday of Columbus Day, which honors Genovese-born explorer Christopher Columbus. Some people reject celebrating him, saying that he represents "the violent history of the colonization in the Western Hemisphere".[4] Indigenous People’s Day was instituted in Berkeley, California, in 1992, to coincide with the 500th anniversary of the arrival of Columbus in the Americas on October 12, 1492. Two years later, Santa Cruz, California, instituted the holiday.[5] Starting in 2014, many other cities and states adopted the holiday.[6]

• October 17

Room: 2-147

Guillaume Remy
(IAS, Princeton)

Modular transformation of conformal blocks via Liouville CFT

Abstract: Conformal blocks are objects of fundamental importance in mathematical physics. They are a key input to the conformal bootstrap program to exactly solve 2D conformal field theory (CFT) and are related to 4D supersymmetric gauge theory through the Alday-Gaiotto-Tachikawa correspondence. They are typically defined as power series via the representation theory of the Virasoro algebra but in this talk we will provide novel probabilistic expressions using the Gaussian free field. This will allow us to obtain many analytic properties such as modular transformations. Our methods are based on recent developments in the probabilistic construction of the Liouville CFT, a theory first introduced to describe random surfaces by A. Polyakov in the context of string theory. Based on joint work with Promit Ghosal, Xin Sun, and Yi Sun.

• October 24

Room: 2-147

Hao Shen

Stochastic Yang-Mills process in 2D and 3D.

Abstract: We will discuss stochastic quantization of the Yang-Mills model on two and three dimensional torus. In stochastic quantization we consider the Langevin dynamic for the Yang-Mills model which is described by a stochastic PDE. We construct local solution to this SPDE and prove that the solution has a gauge invariant property in law, which then defines a Markov process on the space of gauge orbits. We will also describe the construction of this orbit space, on which we have well-defined holonomies and Wilson loop observables. Based on joint work with Ajay Chandra, Ilya Chevyrev, and Martin Hairer.

• October 31

Room: 2-147

Robert Hough
Stony Brook University

Covering systems of congruences

Abstract:

A distinct covering system of congruences is a list of congruences $a_i \bmod m_i, \qquad i = 1, 2, ..., k$ whose union is the integers. Erdős asked if the least modulus $m_1$ of a distinct covering system of congruences can be arbitrarily large ( the minimum modulus problem for covering systems, $1000) and if there exist distinct covering systems of congruences all of whose moduli are odd (the odd problem for covering systems,$25). I'll discuss my proof of a negative answer to the minimum modulus problem, and a quantitative refinement with Pace Nielsen that proves that any distinct covering system of congruences has a modulus divisible by either 2 or 3. The proofs use the probabilistic method and in particular use a sequence of pseudorandom probability measures adapted to the covering process. Time permitting, I may briefly discuss a reformulation of our method due to Balister, Bollobás, Morris, Sahasrabudhe and Tiba which solves a conjecture of Shinzel (any distinct covering system of congruences has one modulus that divides another) and gives a negative answer to the square-free version of the odd problem.

• November 7

Room: 2-147

Infinite cycles in the interchange process in five dimensions

Abstract: In the interchange process on a graph G=(V,E), there is a particle on each vertex of the graph and an independent Poisson clock on each one of the edges. Once the clock of an edge rings, the two particles on the two sides of the edge switch. After time t, the particles are permuted according to a random permutation \pi_t:V\to V. A famous conjecture of Balint Toth states that the following holds when G=\mathbb Z^d :

(1) If d=2, then the permutation \pi_t contains only finite cycles for all t>0.
(2) If d\ge 3, then there exists t_c such that for tt_c, \pi_t contains infinite cycles.

We prove the existence of infinite cycles for all d\ge 5 and all t sufficiently large. To this end, we study the cyclic time random walk obtained by exposing the cycle of the origin in \pi_t. We show that this walk is diffusive using a multi-scale inductive argument.

This is a joint work with Allan Sly.

• November 14

Room: 2-147

Sven Wang
MIT

Minimax density estimation via measure transport

Abstract: This talk is based on the preprint (joint with Y. Marzouk, MIT) https://arxiv.org/abs/2207.10231.

A natural way to represent complex probability measures is to couple them with a known reference distribution via some transport map. This principle has been used in a number of popular recent machine learning methods for high-dimensional inference and generative modeling. In this talk, we study nonparametric density estimators which arise from measure transport. Given a smooth reference distribution and random samples from the unknown density, the estimators are given as the pushforward of a transport map minimizing the empirical Kullback-Leibler risk, or a penalized version thereof. For triangular Knothe-Rosenblatt transports on the d-dimensional unit cube, it is shown that both the non-penalized and penalized versions achieve minimax optimal convergence rates over $H\"older$ classes of densities. The same is proved for sieved wavelet estimators. Our results follow from a concentration inequality for general penalized measure-transport estimators, proved using techniques from M-estimation developed in the late 90's. The minimax rates for triangular maps are then derived using natural anisotropic regularity estimates which we study.

• November 21

Room: 2-147

Emma Bailey
City University of New York (CUNY)

Large deviation estimates for Selberg’s central limit theorem and applications

Abstract: Selberg’s central limit theorem gives that the logarithm of the Riemann zeta function taken at a uniformly drawn height in $[T, 2T]$ behaves as a complex centered Gaussian random variable with variance $\log\log T$. A natural question is to investigate how far the Gaussian decay persists. We present results on the right tail for the real part of the logarithm, where the absolute value of zeta is `unusually large’, on the scale of the exponential of the variance. Our proof employs a recursive scheme of Arguin, Bourgade and Radziwi{\l}{\l} to inductively work with the logarithm of zeta, interpreted as a random walk. The result is in agreement with the corresponding (known) random matrix result, under the usual dictionary, and has a number of corollaries. This work is joint with Louis-Pierre Arguin.

• November 28

Room: 2-147

Sayan Das
Columbia University

Superdiffusivity and localization of continuous polymers

Abstract: We study the Continuum Directed Random Polymer (CDRP) which arises as a universal scaling limit of discrete directed polymers. In this talk, I will present some of the recent progress in understanding the geometry of the CDRP. In particular, I will show CDRP exhibits pointwise localization and pathwise tightness under 2/3 scaling, confirming the existing predictions in polymer literature under continuous setting. I will explain how our results also shed light on certain properties of the KPZ equation (free energy of the CDRP) such as ergodicity and limiting Bessel behaviors around the maximum. This is based on two joint works with Weitao Zhu.

• December 5

Room: 2-147

Changji Xu
Harvard University

Spectral gap estimates for mixed p-spin models at high temperature

Abstract: We consider general mixed p-spin mean field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form corresponding to Glauber dynamics is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the N-spin system to that of suitably conditioned subsystems.​​

• December 12

Room: 2-147

Hoi Nguyen
The Ohio State University

On roots of random trigonometric polynomials and related models

Abstract: In this talk we will discuss various basic statistics of the number of real roots of random trigonometric polynomials, as well as the minimum modulus and the nearest roots statistics to the unit circle of Kac polynomials. We emphasize the universality aspects of all these problems.

Based on joint works with Cook, Do, O. Nguyen, Yakir and Zeitouni